(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic geometry study.

(2) I'm guessing a line is the variety $\mathcal{V}(f_1,\ldots,f_{n-1})$, where $f_i$ are linear homogenous polynomials, whose coefficients form a $n\times n\!-\!1$ matrix of full rank. Yes or no? Another guess would be, that a line in $\mathbb{P}^n(k)$ is uniquely determined by two points $a\!=\![a_0\!:\!\ldots\!:\!a_n]$, $b\!=\![b_0\!:\!\ldots\!:\!b_n]$, such that the matrix $\begin{bmatrix} a \\ b \end{bmatrix}$ is of rank $2$. But how is such a line parametrized? Is any of my two attempts of a definition correct?

(3) What are the defining equations of two intersecting lines in $\mathbb{P}^3$? And now, most importantly: how can I compute the Hilbert polynomial of such a variety?

For such an elementary concept, one would expect it to be the first object defined, but to my annoyance and frustration, I have yet to see an official definition. I have the book Introduction to Algebraic Geometry (Hassett), as well as Algebraic Curves (Fulton) as my main source. Any references would be highly desirable.

thank you


Regarding the third part of your question: The intersection of two lines in $\mathbb P^3$ is generically empty (if you write down two random lines, they will be skew), but sometimes the lines will be coplanar (i.e. lie in a common plane), and then they will meet in a point (as any two lines in $\mathbb P^2$ do).

If you know that a priori that the two lines meet in a point, then they must be coplanar, and the problem is the same as for two intersecting lines in $\mathbb P^2$.

  • $\begingroup$ How does one know that the problem is the same as for two intersecting lines in $\mathbb{P}^2$? $\endgroup$ – Leo Mar 1 '11 at 16:20
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    $\begingroup$ Aha: Hassett p. 214 property 2: the Hilbert polynomial is invariant under projectivities, i.e. $HP_\mathcal{V}=HP_{A(\mathcal{V})}$ for an invertible $(n+1)\times(n+1)$ matrix $A$ over $k$. Right. But how do I calculate $HP$ for two intersecting lines in $\mathbb{P}^2$? $\endgroup$ – Leo Mar 1 '11 at 16:35
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    $\begingroup$ @Leon: Dear Leon, What techniques do you know for computing Hilbert polynomials? Regards, $\endgroup$ – Matt E Mar 1 '11 at 18:56
  • $\begingroup$ basically what's in Hassett p. 214: $HP_{\mathcal{V}(f)}(t)=\binom{t+n}{n}-\binom{t-d+n}{n}$ for $f$ homogenous in $n+1$ variables of degree $d$; $HP_S(t)=|S|$ for $S$ finite; $HP_\mathcal{V}$ is invariant under projectivities; $HP_{\mathcal{V}(f,g)}=deg(f)deg(g)$ for $f,g$ homogenous in $2+1$ variables with no common factors; $HP_{\cup_{i=1}^k\mathcal{V}_i}(t)=\sum_{i=1}^k HP_{\mathcal{V}_i}(t)$ for $\mathcal{V}_i$ pairwise disjoint and $k$ alg. closed. $\endgroup$ – Leo Mar 1 '11 at 19:23
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    $\begingroup$ @Leon: Then why don't you find a homoogeneous polynomial describing two lines in $\mathbb P^2$ and apply the definition? Regards, $\endgroup$ – Matt E Mar 1 '11 at 20:51

The line through $(a_0 : ... : a_n)$ and $(b_0 : ... : b_n)$ is uniquely parameterized by $(a_0 X + b_0 Y : a_1 X + b_1 Y : ... a_n X + b_n Y)$. Note that this precisely describes a morphism $\mathbb{P}^1 \to \mathbb{P}^n$ which is an isomorphism onto its image. I am surprised this is not given somewhere in Fulton.

  • $\begingroup$ Hmm, you are right: Fulton section 4, page 48, exercise 4.13. Sorry, I didn't expect it to be in the excercises, and so late in the section. $\endgroup$ – Leo Mar 1 '11 at 2:44

(2) Your guess is correct, except I think it is an $(n-1)\times (n+1)$ matrix. Think of your line as the intersection of $n-1$ hyperplanes in non-degenerate position.

(3) Assuming you meant the union of two lines $L_1, L_2$, then the defining variety is given by $I_1I_2$.

As for the Hilbert poly., use:

$$0 \to \mathcal O_{L_1\cup L_2} \to \mathcal O_{L_1}\oplus \mathcal O_{L_2} \to \mathcal O_{L_1\cap L_2} \to 0$$

and the fact that Hilbert poly. are additive.

Details added: the two terms on the right of the above sequence can be calculated easily. Think of a line as $Proj(k[x,y])$ and a point (since you know $L_1,L_2$ intersect at a point) as $Proj(k[x])$. I am sure you can find the Hilbert polynomials of those rings.

  • $\begingroup$ (2) yes, of course, $(n-1)\times(n+1)$. $\endgroup$ – Leo Mar 1 '11 at 15:03
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    $\begingroup$ (3) Umm, I'm not sure, what you are suggesting. All I know of the Hilbert function and Hilbert polynomial is from Hasset, section 12. Let $L_1, L_2$ be two lines in $\mathbb{P}^3$ with $L_1\cap L_2=\{P\}$. How does one calculate $HP_{L_1\cup L_2=:\mathcal{V}}(t)=HP_{R(\mathcal{V})}(t)=HP_{k[x_0:\ldots:x_3]/I(\mathcal{V})}(t)$? Could you point me to any specific theorem/lemma? Preferrably in Hasset $\endgroup$ – Leo Mar 1 '11 at 16:18
  • $\begingroup$ @Leon Lampret: I added some details. $\endgroup$ – curious Mar 1 '11 at 18:43
  • $\begingroup$ Hmm, I am not familiar with what you are talking about. Well, since my question has been answered in one way, there is no need to bother and try to explain it to me, but if you do have the patience and a little time, would you care to tell me the exact formulation of this additive property of Hilbert polynomials? Regards, $\endgroup$ – Leo Mar 2 '11 at 18:05
  • $\begingroup$ @curious: Of course it is obvious for $L_1 \cap L_2$ a point, but why this sequence is exact in the right term in general? $\endgroup$ – evgeny Jun 9 '14 at 18:19

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